of_basis_left_of_commute — Mathlib

English

If a basis of A is linearly independent relative to B and all pairs commute, then A and B are linearly disjoint.

Русский

Если базис A линейно независим относительно B и все пары компутивны, тогда A и B линейно несовместны.

LaTeX

$$$(of\\_basis\\_left\\_of\\_commute)$$$

Lean4

/-- If a basis of `A` is also `B`-linearly independent, if elements in `A` and `B` commute,
then `A` and `B` are linearly disjoint. -/
theorem of_basis_left_of_commute {ι : Type*} (a : Basis ι R A) (H : LinearIndependent B (A.val ∘ a))
    (hc : ∀ (a : A) (b : B), Commute a.1 b.1) : A.LinearDisjoint B :=
  (of_basis_right B A a H).symm_of_commute fun _ _ ↦ (hc _ _).symm

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